NEWTON POLYGONS AND FAMILIES OF POLYNOMIALS

Niveau: Secondaire, Lycée, Première
NEWTON POLYGONS AND FAMILIES OF POLYNOMIALS ARNAUD BODIN Abstract. We consider a continuous family (fs), s ? [0, 1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant. We firstly prove that the set of critical values at infinity depends continuously on s, and secondly that the degree of the fs is constant (up to an algebraic automorphism of C2). 1. Introduction We consider a family (fs)s?[0,1] of complex polynomials in two variables with isolated singularities. We suppose that coefficients are continuous func- tions of s. For all s, there exists a finite bifurcation set B(s) such that the restriction of fs above C \ B(s) is a locally trivial fibration. It is known that B(s) = Baff (s) ? B∞(s), where Baff (s) is the set of affine critical val- ues, that is to say the image by fs of the critical points; B∞(s) is the set of critical values at infinity. For c /? B(s), the Euler characteristic verifies ?(f?1s (c)) = µ(s) + ?(s), where µ(s) is the affine Milnor number and ?(s) is the Milnor number at infinity.

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